By Stephan Körner

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C) qu’on a une identification cosqm ◦ cosqn = cosqn pour m ≥ n. La fl`eche cosqn+1 (X) → cosqn (X) s’interpr`ete alors comme une fl`eche ˜ τ : cosqn+1 (X) → cosqn+1 (X) ˜ = cosqn (X). Remarquons que τp est un isomorphisme si p ≤ n, cette fl`eche s’identifiant o` u l’on a pos´e X a l’identit´e de ` ˜ p=X ˜ p = cosqn (X)p = Xp . cosqn+1 (X)p = Xp → cosqn+1 (X) Le morphisme τn+1 s’identifie quant `a lui `a la fl`eche canonique Xn+1 → cosqn (X)n+1 . 2. — Soit n un entier ≥ −1 et τ ∈ HomS (X, X). phisme si p ≤ n et τn+1 est un morphisme de descente cohomologique universelle.

Phisme si p ≤ n et τn+1 est un morphisme de descente cohomologique universelle. Alors, pour tout p, la fl`eche ˜ p cosqn+1 (X)p → cosqn+1 (X) est un morphisme de descente cohomologique universelle. D´emonstration. — On peut supposer p > n + 1. On ´ecrit alors cosqn+1 (X)p = lim Xq ←− [q]→[p] q≤n+1 comme le noyau de la double fl`eche ΠX = d´ ef Xq ⇒ Xi = ΞX α [q]→[p] q≤n+1 [i] →[j] [p] j≤n+1 o` u la composante αX d’indice α ∈ Hom[p] ([i], [j]) de la double fl`eche est la double fl`eche form´ee d’une part du morphisme ΠX → Xi de projection d’indice [i] → [p] et, d’autre part, du morphisme ΠX → Xj → Xi , compos´e de la projection d’indice [j] → [p] et de α ∈ Hom(Xj , Xi ).

4) sont de descente cohomologique. 5), π et π ˜ sont des ´equivalences de S-descente cohomologiques. Passons `a f˜. 3), il suffit de prouver que les morphismes d’espaces simpliciaux obtenus en regardant chaque ligne d’indice p ≥ 0 sont des ´equivalences. Ce morphisme est la premi`ere projection ϕp : (Y• )p+1 → (Y• )p o` u le produit est pris sur Y. 2), (Y• )p+1 et (Y• )p sont leur propre n + 1-cosquelettes. Par ailleurs, comme Y et Y co¨ıncident en degr´e ≤ n, le morphisme sqn (φp ) est un isomorphisme.

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Categorial frameworks by Stephan Körner


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